METHODS OF INTERPOL A.TIO]Sr. 351 



Similarly;, the formula — 



^ I 7^1(3 + 5 {1(2 + ^^4) - (wi 4- W5) I 



will give exact results for any series of equidistant values of the func- 

 tioii— 



provided that these values are taken at intervals equal to unity. So, 

 also — 



^3 = ^^^r4 I ^ "3 + 3 («2 + W4) — (Wi + «5) I 

 will be exact for equidistant values of — 



^ 7: X ^ ttX 



u = A 4- Bx -{- C sm ^T- +D cos -rr 



taken at intervals of unity. But these two last adjustment-formulas 

 will not apply to series of values of the functions taken at any equal 

 intervals other than unity, and therein lies their inferiority to the for- 

 mulas of our second method, which apply alike to all equidistant values of 

 the algebraic function independently of the magnitude of the interval. 



We may notice further that the circular function — 



u = Bi sin {x0) 4- Ci cos {x6) -f Bo sin 2 {xO) -f C2 cos 2 {x6) -f 



+ Bm sin m (xO) -\- Cm cos m [xO] 



is a recurring function of the 2wth order, whose equation of relation 

 has no real roots, but has m pairs of imaginary ones, such as to make y, 

 Yi^ Y2-, &c., all equal to unity, the equation of relation being a reciproca) 

 equation. The property demonstrated by Airy, in connection with a 

 process for smoothing down the irregularities of series of circular form, 

 in the Transactions of the Eoyal Society for 1870, is a consequence of 

 this recurrent character of the function. If a constant term A were 

 added to the function, the series would be raised to the (2 m + l)th order, 

 the constant being represented in the equation of relation by a single 

 real root equal to unity. 



April, 1874. 



APPENDIX. 

 f 



Additions to Table III. 



When these adjustment-formulas are to include a considerable numbei 

 of terms, the easiest way to construct them will be to regard the weights, 

 in their fractional form, as ordinates to a curve of the tenth degree, and 

 to take the weight of the middle term as the axis of Y, so that the curve 

 will be symmetrical on either side of this axis, and its equation, con- 

 taining only even powers of r», may be written— 



2/ = A + B ^2 _^ C ^4 + D *6 + E x"^ + F i»io 



Suppose, for instance, that we wish to find the 23 term formula. Lei 



